So the thing is that while Cp and Cpk are technically in the formula, in practice they end up cancelling themselves out and the results are only dependent on sigma-within and the difference between the target value and the mean. It isn't affected at all by the spec limits themselves.

I tried walking through the math by hand and it looks like the USL and LSL don't show up in the final formula, but to be honest I am not the best at this kind of math and not super sure how to write it out here meaningfully anyways.

But just to show what I mean, take a look here

So in this case my system has a T = 0 with spec at +/- 2. The mean is at 0.03, and the PPK/CPK are both beautiful on this, but the target index is flagging.

Alternatively take a look at this. Same T same USL/LSL

In this case the Target Index is better (not great, but better), but the Mean is much much farther off target.

The reason why is because of the Sigma-Within in the first case is extremely narrow, wherease it's much worse in the second case.

Further proof. Let's change the USL/LSL size to +/- 3 for those two

Here is the first

Here is the second

Note that the PPK and CPK both improve, as would be expected since the spec window has opened up farther. Yet the Target Index did not change at all.

Therefore: The target index is independent of the spec window (assuming that the Target = (USL + LSL/2))

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This isn't to say that the Target Index isn't useful. But really you're effectively just z-scoring your data. How many sigma-withins is your mean away from your target. In my case the target is 0, so the Mean = x-bar.

So by that logic TI = Mean / Sigma-Within So in the TI = 1.58 case I have 0.0353/0.01933 = 1.5794

For the second case TI = Mean/Sigma-Within = 0.6648

Now if your target <> 0 then you would have to subtract the Target from the Mean. But either way USL and LSL are not part of the equation.

Edit: If anyone wants to test just use the Semiconductor Capability sample set. Change the USL/LSL however you want. As far as I can tell it doesn't affect the TI.